Theorem summodclem2 10990

Description: Lemma for summodc 10991. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)

Hypotheses

Ref	Expression
isummo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
isummo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
summodclem2.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))

Assertion

Ref	Expression
summodclem2	⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))

Distinct variable groups:   𝑘,𝑛,𝐴   𝑛,𝐹   𝜑,𝑘,𝑛   𝐴,𝑓,𝑗,𝑚,𝑘,𝑛   𝐵,𝑛   𝑓,𝐹,𝑘,𝑚   𝜑,𝑓,𝑚   𝑥,𝑓,𝑘,𝑚,𝑛   𝑦,𝑓,𝑚

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑗)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚)   𝐹(𝑥,𝑦,𝑗)   𝐺(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚,𝑛)

Proof of Theorem summodclem2

Dummy variables 𝑎 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5353	. . . . 5 ⊢ (𝑚 = 𝑎 → (ℤ≥‘𝑚) = (ℤ≥‘𝑎))
2	1	sseq2d 3077	. . . 4 ⊢ (𝑚 = 𝑎 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐴 ⊆ (ℤ≥‘𝑎)))
3	1	raleqdv 2590	. . . 4 ⊢ (𝑚 = 𝑎 → (∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴))
4		seqeq1 10062	. . . . 5 ⊢ (𝑚 = 𝑎 → seq𝑚( + , 𝐹) = seq𝑎( + , 𝐹))
5	4	breq1d 3885	. . . 4 ⊢ (𝑚 = 𝑎 → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑎( + , 𝐹) ⇝ 𝑥))
6	2, 3, 5	3anbi123d 1258	. . 3 ⊢ (𝑚 = 𝑎 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)))
7	6	cbvrexv 2613	. 2 ⊢ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥))
8		simplr3 993	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → seq𝑎( + , 𝐹) ⇝ 𝑥)
9		simplr1 991	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ⊆ (ℤ≥‘𝑎))
10		uzssz 9195	. . . . . . . . . . . 12 ⊢ (ℤ≥‘𝑎) ⊆ ℤ
11	9, 10	syl6ss 3059	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ⊆ ℤ)
12		1zzd 8933	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 1 ∈ ℤ)
13		simprl 501	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑚 ∈ ℕ)
14	13	nnzd 9024	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑚 ∈ ℤ)
15	12, 14	fzfigd 10045	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (1...𝑚) ∈ Fin)
16		simprr 502	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
17		f1oeng 6581	. . . . . . . . . . . . . 14 ⊢ (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (1...𝑚) ≈ 𝐴)
18	15, 16, 17	syl2anc 406	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (1...𝑚) ≈ 𝐴)
19	18	ensymd 6607	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ≈ (1...𝑚))
20		enfii 6697	. . . . . . . . . . . 12 ⊢ (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
21	15, 19, 20	syl2anc 406	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ∈ Fin)
22		zfz1iso 10425	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
23	11, 21, 22	syl2anc 406	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
24		isummo.1	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
25		simplll 503	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
26		isummo.2	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
27	25, 26	sylan 279	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
28		eleq1w 2160	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
29	28	dcbid 792	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑘 ∈ 𝐴))
30		simpr2 956	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴)
31	30	ad2antrr 475	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑎)) → ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴)
32		simpr 109	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑎)) → 𝑘 ∈ (ℤ≥‘𝑎))
33	29, 31, 32	rspcdva 2749	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑎)) → DECID 𝑘 ∈ 𝐴)
34		summodclem2.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
35		eqid 2100	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑔‘𝑛) / 𝑘⦌𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑔‘𝑛) / 𝑘⦌𝐵, 0))
36		simprll 507	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
37		simpllr 504	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑎 ∈ ℤ)
38		simplr1 991	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ≥‘𝑎))
39		simprlr 508	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
40		simprr 502	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
41	24, 27, 33, 34, 35, 36, 37, 38, 39, 40	summodclem2a 10989	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
42	41	expr 370	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
43	42	exlimdv 1758	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
44	23, 43	mpd 13	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
45		climuni 10901	. . . . . . . . 9 ⊢ ((seq𝑎( + , 𝐹) ⇝ 𝑥 ∧ seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
46	8, 44, 45	syl2anc 406	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
47	46	anassrs 395	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
48		eqeq2 2109	. . . . . . 7 ⊢ (𝑦 = (seq1( + , 𝐺)‘𝑚) → (𝑥 = 𝑦 ↔ 𝑥 = (seq1( + , 𝐺)‘𝑚)))
49	47, 48	syl5ibrcom 156	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑦 = (seq1( + , 𝐺)‘𝑚) → 𝑥 = 𝑦))
50	49	expimpd 358	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
51	50	exlimdv 1758	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
52	51	rexlimdva 2508	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
53	52	r19.29an 2532	. 2 ⊢ ((𝜑 ∧ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
54	7, 53	sylan2b 283	1 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 103  DECID wdc 786   ∧ w3a 930   = wceq 1299  ∃wex 1436   ∈ wcel 1448  ∀wral 2375  ∃wrex 2376  ⦋csb 2955   ⊆ wss 3021  ifcif 3421   class class class wbr 3875   ↦ cmpt 3929  –1-1-onto→wf1o 5058  ‘cfv 5059   Isom wiso 5060  (class class class)co 5706   ≈ cen 6562  Fincfn 6564  ℂcc 7498  0cc0 7500  1c1 7501   + caddc 7503   < clt 7672   ≤ cle 7673  ℕcn 8578  ℤcz 8906  ℤ_≥cuz 9176  ...cfz 9631  seqcseq 10059  ♯chash 10362   ⇝ cli 10886

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-isom 5068  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-frec 6218  df-1o 6243  df-oadd 6247  df-er 6359  df-en 6565  df-dom 6566  df-fin 6567  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-fz 9632  df-fzo 9761  df-seqfrec 10060  df-exp 10134  df-ihash 10363  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-clim 10887

This theorem is referenced by:  summodc  10991